1 info 2950 prof. carla gomes [email protected] module basic structures: functions and sequences...
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INFO 2950
Prof. Carla [email protected]
Module Basic Structures: Functions and Sequences
Rosen 2.3 and 2.4
Functions
Suppose we have:
How do you describe the yellow function?
What’s a function ? f(x) = -(1/2)x – 1/2
x
f(x)
Functions
More generally:
Definition:Given A and B, nonempty sets, a function f from A to B is an assignment of exactly one element of B to each element of A. We write f(a)=b if b is the element of B assigned by function f to the element a of A.If f is a function from A to B, we write f : AB.
Note: Functions are also called mappings or transformations.
B
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Functions
A = {Michael, Toby , John , Chris , Brad }B = { Kathy, Carla, Mary}
Let f: A B be defined as f(a) = mother(a).
Michael Toby
John Chris Brad
Kathy
Carol
Mary
A B
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Functions
More generally:
f: RR, f(x) = -(1/2)x – 1/2
domain co-domain
A - Domain of f B- Co-Domain of f
B
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Functions
More formally: a function f : A B is a subset of AxB where
a A, ! b B and <a,b> f. (note: ! for unique exists.)
A
B
A
B
a point!
a collection of points!
Why not?
Functions - image & preimage
For any set S A, image(S) = {b : a S, f(a) = b}
So, image({Michael, Toby}) = {Kathy} image(A) = B - {Carol}
image(S)
image(John) = {Kathy} pre-image(Kathy) = {John, Toby, Michael}
range of f image(A)
Michael Toby
John Chris Brad
Kathy
Carol
Mary
A B
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Functions - injection
A function f: A B is one-to-one (injective, an injection) if a,b,c, (f(a) = b f(c) = b) a = c
Not one-to-one
Every b B has at most 1 preimage.
Michael Toby
John Chris Brad
Kathy
Carol
Mary
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Functions - surjection
A function f: A B is onto (surjective, a surjection) if b B, a A f(a) = b
Not onto
Every b B has at least 1 preimage.
Michael Toby
John Chris Brad
Kathy
Carol
Mary
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Functions – one-to-one-correspondence
or bijection
A function f: A B is bijective if it is one-to-one and onto.
Anna Mark John Paul
Sarah
Carol
Jo
Martha
Dawn
Eve
Every b B has exactly 1 preimage.
An important implication of this
characteristic:The preimage (f-1)
is a function!They are
invertible.
Anna Mark John
Paul
Sarah
Carol Jo Martha Dawn
Eve
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Functions: inverse function
Definition:
Given f, a one-to-one correspondence from set A to set B, the inverse
function of f is the function that assigns to an element b belonging to B the unique element a in A such that f(a)=b. The inverse function is denoted f-1 . f-1 (b)=a, when f(a)=b.
B
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Functions - examples
Suppose f: R+ R+, f(x) = x2.
Is f one-to-one?Is f onto?Is f bijective?
yes
yesyes
This function is invertible.
here
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Functions - examples
Suppose f: R R+, f(x) = x2.
Is f one-to-one?Is f onto?Is f bijective?
noyes
no
This function is not invertible.
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Functions - examples
Suppose f: R R, f(x) = x2.
Is f one-to-one?Is f onto?Is f bijective?
no
no
no
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Functions - composition
Let f: AB, and g: BC be functions. Then the composition of f and g is:
(f o g)(x) = f(g(x))
Note: (f o g) cannot be defined unless the range of g is a subset of the domain of f.
“f composed with g”
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Example:
Let f(x) = 2 x +3; g(x) = 3 x + 2;
(f o g) (x) = f(3x + 2) = 2 (3 x + 2 ) + 3 = 6 x + 7.
(g o f ) (x) = g (2 x + 3) = 3 (2 x + 3) + 2 = 6 x + 11.
As this example shows, (f o g) and (g o f) are not necessarily equal – i.e, the composition of functions is not commutative.
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Note:
(f -1 o f) (a) = f -1(f(a)) = f -1(b) = a.
(f o f -1) (b) = f (f -1(b)) = f-(a) = b.
Therefore (f-1o f ) = IA and (f o f-1) = IB where IA and IB are the identity
function on the sets A and B. (f -1) -1= f
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Some important functions
Absolute value:
Domain R; Co-Domain = {0} R+
|x| = x if x ≥0-x if x < 0
Ex: |-3| = 3; |3| = 3
Floor function (or greatest integer function): Domain = R; Co-Domain = Z
x = largest integer not greater than x
Ex: 3.2 = 3; -2.5 =-3
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Some important functions
Ceiling function: Domain = R; Co-Domain = Z
x = smallest integer greater than or equal to x
Ex: 3.2 = 4; -2.5 =-2
≤
≤
≤
≤
+
+
+
++
+
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Some important functions
Factorial function: Domain = Range = N
n! = n (n-1)(n-2) …, 3 x 2 x 1Ex: 5! = 5 x 4 x 3 x 2 x 1 = 120
Note: 0! = 1 by convention.
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Mod (or remainder): Domain = N x N+ = {(m,n)| m N, n N+ }Co-domain Range = N
m mod n = m - m/n n
Ex: 8 mod 3 = 8 - 8/3 3 = 2 57 mod 12 = 9;
Note: This function computes the remainder when m is divided by n.The name of this function is an abbreviation of m modulo n.
Note also that this function is an example in which the domain of the function is a 2-tuple.
(*: also valid for negative numbers; but here we only consider natural numbers)
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Exponential Function
Exponential function: Domain = R+ x R = {(a,x)| a R+, x R }Co-domain Range = R+
f(x) = a x
Note: a is a positive constant; x varies.
Ex: f(n) = a n = a x a …, x a (n times)
How do we define f(x) if x is not a positive integer?
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Exponential function: How do we define f(x) if x is not a positive integer?
Important properties of exponential functions:
(1) a (x+y) = ax ay; (2) a1 = a; (3) a0 = 1
See:
)(
...
;
;12123
11112
timesnaaa
aaaaaaa
aaaaaa
n
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We get:
aathereforeaaaaaa
aathereforeaaaa
athereforeaaaaabbbbbb
2
122
1
2
1
2
1
2
1
2
11
)(0
00011
)(
11
1
By similar arguments:
mnmnn
mmxxxmx
kk
aaathereforeatimesmaaa
aa
)()(,)()(1
1
Note: This determines ax for all x rational. x is irrational by continuity (we’ll skip “details”).
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Some important functions:Logarithm Function
Logarithm base a: Domain = R+ x R = {(a,x)| a R+, a>1, x R }Co-domain / Range = R
y = log a (x) ay = x Ex: log 2 (8) =3; log 2 (16) =4; 3 < log 2 (15) <4.
Key properties of the log function (they follow from those for exponential):
1. log a (1) = 0 (because a0 =1)2. log a (a) = 1 (because a1 =a)3. log a (xy) = log a (x) + log a (y) (similar arguments)4. log a (xr) = r log a (x)
5. log a (1/x) = - log a (x) (note 1/x = x-1)6. log b (x) = log a (x) / log a (b)
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Examples:
log 2 (1/4) = - log 2 (4)
= - 2.
log 2 (-4) undefined
log 2 (210 35 )= log 2 (210) + log 2 (35 )
= 10 log 2 (2) + 5log 2 (3 )
= 10 + 5 log 2 (3 )
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Limit Properties of Log Function
0)log(lim
)log(lim
xx
x
xx
As x gets large, log(x) grows without bound. But x grows MUCH faster than log(x)…more soon on growth rates.
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Some important functions:Polynomials
Polynomial function: Domain = usually R Co-domain Range = usually R
Pn(x) = anxn + an-1xn-1 + … + a1x1 + a0
n, a nonnegative integer is the degree of the polynomial;an 0 (so that the term anxn actually appears)
(an, an-1, …, a1, a0) are the coefficients of the polynomial.
Ex: y = P1(x) = a1x1 + a0 linear functiony = P2(x) = a2x2 + a1x1 + a0 quadratic polynomial or function
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Exponentials grow MUCH faster than polynomials:
10lim 0
bif
b
xaax
kk
x
We’ll talk more about growth rates in the next module….
Sequences
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Sequences
Definition:A sequence {ai} is a function f: A N {0} S, where we write
ai to indicate f(i). We call ai term I of the sequence.
Examples:
Sequence {ai}, where ai = i is just a0 = 0, a1 = 1, a2 = 2, …
Sequence {ai}, where ai = i2 is just a0 = 0, a1 = 1, a2 = 4, …
Sequences of the form a1, a2, …, an are often used in computer science.
(always check whether sequence starts at a0 or a1)
These finite sequences are also called strings. The length of a string is the number ofterms in the string. The empty string, denoted by , is the string that has no terms.
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Geometric and Arithmetic Progressions
Definition: A geometric progression is a sequence of the form
,,,,,, 32 narararara
The initial term a and the common ratio r are real numbers
Definition: An arithmetic progression is a sequence of the form
,,,3,2,, ndadadadaa
The initial term a and the common difference d are real numbers
Note: An arithmetic progression is a discrete analogue of the linear function f(x) = dx + a
Notice differences in growth rate.
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Summation
The symbol (Greek letter sigma) is used to denote summation.
The limit:
ai
i1
k
a1 a2 ak
ai
i1
limn
ai
i1
n
i is the index of the summation, and the choice of letter i is arbitrary;
the index of the summation runs through all integers, with its lower limit 1and ending upper limit k.
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The laws for arithmetic apply to summations
cai bi i1
k
c ai
i1
k
bi
i1
k
Use associativity to separate the b terms from the a terms.
Use distributivity to factor the c’s.
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Summations you should know…
What is S = 1 + 2 + 3 + … + n?
You get n copies of (n+1). But we’ve over added by a factor of 2. So just divide by 2.
S = 1 + 2 + … + n
S = n + n-1 + … + 1
2s = n+1 + n+1 + … + n+1
Write the sum.
Write it again.
Add together.
kk1
n
n(n 1)
2
(little) Gauss in 4th grade.
Why whole number?
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What is S = 1 + 3 + 5 + … + (2n - 1)?
Sum of first n odds.
(2k 1)k1
n
2 kk1
n
1k1
n
2n(n 1)
2
n
n2
Pleasantly simple expression
What is S = 1 + 3 + 5 + … + (2n - 1)?
Sum of first n odds.
n2
The visual way.
What is S = 1 + r + r2 + … + rn
Geometric Series
rk
k0
n
1 r rn
r rk
k0
n
r r2 rn1
Multiply by r
Subtract the summations
rk
k0
n
r rk
k0
n
1 rn1
factor
(1 r) rk
k0
n
1 rn1divide
rk
k0
n
1 rn1
(1 r)DONE!
What about:
rk
k0
1 r rn
nlim
1 rn1
(1 r)
If r 1 this blows up.
If r < 1 we can say something.
rk
k0
nlim rk
k0
n
1
(1 r)
Try r = ½.
Useful Summations
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Infinite Cardinality
How can we extend the notion of cardinality to infinite sets?
Definition: Two sets A and B have the same cardinality if and only if there exists a bijection (or a one-to-one correspondence) between them, A ~ B.
We split infinite sets into two groups:
1. Sets with the same cardinality as the set of natural numbers2. Sets with different cardinality as the set of natural numbers
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Infinite Cardinality
Definition: A set is countable if it is finite or has the same cardinality as the set of positive integers.
Definition: A set is uncountable if it is not countable.
Definition: The cardinality of an infinite set S that is countable is denotes by 0א (where א is aleph, the first letter of the Hebrew alphabet). We write |S| = 0א and say that S has cardinality “aleph null”.
Note: Georg Cantor defined the notion of cardinality and was the first to realize that infinite sets can
have different cardinalities. 0א is the cardinality of the natural numbers; the next larger cardinality
is aleph-one 1א , then, 2א and so on.
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Infinite Cardinality:Odd Positive Integers
2
Example: The set of odd positive integers is a countable set.
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Infinite Cardinality:Odd Positive Integers
Example: The set of odd positive integers is a countable set.
Let’s define the function f, from Z+ to the set of odd positive numbers,
f(n) = 2 n -1
We have to show that f is both one-to-one and onto.
a) one-to-one
Suppose f(n)= f(m) 2n-1 = 2m-1 n=m
b) onto
Suppose that t is an odd positive integer. Then t is 1 less than an even integer 2k, where k is a natural number. hence t=2k-1= f(k).
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Infinite Cardinality:Odd Positive Integers
2
Example: The set of odd positive integers is a countable set.
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Infinite Cardinality:Integers
Example: The set of integers is a countable set.
Lets consider the sequence of all integers, starting with 0: 0,1,-1,2,-2,….
We can define this sequence as a function:
oddZnn
evenZnn
,2
)1(
,2
f(n) =
Show at home that it’s one-to-one and onto
20 1 -1 2
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Infinite Cardinality:Rational Numbers
Example: The set of positive rational numbers is a countable set.
Hmm…
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Infinite Cardinality:Rational Numbers
Example: The set of positive rational numbers is a countable set
Key aspect to list the rational numbers as a sequence – every positive number is the quotient p/q of two positive integers.
Visualization of the proof.
p+q=2p+q=3
p+q=4 p+q=5 p+q=6
Since all positive rational numbers are listed once, the set of positive rational numbers is countable.
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Uncountable Sets:Cantor's diagonal argument
The set of all infinite sequences of zeros and ones is uncountable.
Consider a sequence, 10,,,,, 21 iin aoranaaa
For example:
So in general we have:
i.e., sn,m is the mth element of the nth sequence on the list.
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It is possible to build a sequence, say s0, in such a way that
its first element is different from the first element of the first sequence in the list,
its second element is different from the second element of the second sequence in
the list, and, in general,
its nth element is different from the nth element of the nth sequence in the list.
In other words, s0,m will be 0 if sm,m is 1,
and s0,m will be 1 if sm,m is 0.
Uncountable Sets:Cantor's diagonal argument
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The sequence s0 is distinct from all the sequences in the list. Why?
Let’s say that s0 is identical to the 100th sequence, therefore, s0,100=s100,100.
In general, if it appeared as the nth sequence on the list, we would have s0,n = sn,n,
which, due to the construction of s0, is impossible.
Note: the diagonal elements are highlighted,showing why this is called the diagonal argument
Uncountable Sets:Cantor's diagonal argument
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From this it follows that the set T, consisting of all infinite sequences of
zeros and ones, cannot be put into a list s1, s2, s3, ... Otherwise, it would
be possible by the above process to construct a sequence s0 which would
both be in T (because it is a sequence of 0's and 1's which is by the
definition of T in T) and at the same time not in T (because we can
deliberately construct it not to be in the list). T, containing all such
sequences, must contain s0, which is just such a sequence. But since s0
does not appear anywhere on the list, T cannot contain s0.
Therefore T cannot be placed in one-to-one correspondence with the
natural numbers. In other words, the set of infinite binary strings is
uncountable.
Uncountable Sets:Cantor's diagonal argument
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Real Numbers
Example; The set of real numbers is an uncountable set.
Let’s assume that the set of real numbers is countable.
Therefore any subset of it is also countable, in particular the interval [0,1].
How many real numbers are in interval [0, 1]?
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Real Numbers
How many real numbers are in interval [0, 1]?
0.4 3 2 9 0 1 3 2 9 8 4 2 0 3 9 …0.8 2 5 9 9 1 3 2 7 2 5 8 9 2 5 …0.9 2 5 3 9 1 5 9 7 4 5 0 6 2 1 …
…
“Countably many! There’s part of the list!”
“Are you sure they’re all there?”
0.5 3 6 …So, the reals are “uncountable.”
Counterexample: Use diagonalization
to create a new numberthat differs in the ith
position of the ith number
by 1.